Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.998 + 0.0527i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.79 − 1.79i)2-s + (−1.66 + 0.491i)3-s + 4.47i·4-s + (1.87 + 1.21i)5-s + (3.87 + 2.10i)6-s + (−0.707 + 0.707i)7-s + (4.45 − 4.45i)8-s + (2.51 − 1.63i)9-s + (−1.20 − 5.56i)10-s + 1.56i·11-s + (−2.19 − 7.43i)12-s + (2.21 + 2.21i)13-s + 2.54·14-s + (−3.71 − 1.08i)15-s − 7.09·16-s + (3.60 + 3.60i)17-s + ⋯
L(s)  = 1  + (−1.27 − 1.27i)2-s + (−0.958 + 0.283i)3-s + 2.23i·4-s + (0.840 + 0.541i)5-s + (1.58 + 0.859i)6-s + (−0.267 + 0.267i)7-s + (1.57 − 1.57i)8-s + (0.839 − 0.543i)9-s + (−0.380 − 1.75i)10-s + 0.472i·11-s + (−0.634 − 2.14i)12-s + (0.615 + 0.615i)13-s + 0.680·14-s + (−0.959 − 0.280i)15-s − 1.77·16-s + (0.874 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.998 + 0.0527i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.998 + 0.0527i)$
$L(1)$  $\approx$  $0.460432 - 0.0121530i$
$L(\frac12)$  $\approx$  $0.460432 - 0.0121530i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (1.66 - 0.491i)T \)
5 \( 1 + (-1.87 - 1.21i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.79 + 1.79i)T + 2iT^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
13 \( 1 + (-2.21 - 2.21i)T + 13iT^{2} \)
17 \( 1 + (-3.60 - 3.60i)T + 17iT^{2} \)
19 \( 1 - 1.68iT - 19T^{2} \)
23 \( 1 + (-0.995 + 0.995i)T - 23iT^{2} \)
29 \( 1 + 8.91T + 29T^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (-0.440 + 0.440i)T - 37iT^{2} \)
41 \( 1 + 6.44iT - 41T^{2} \)
43 \( 1 + (5.47 + 5.47i)T + 43iT^{2} \)
47 \( 1 + (-3.69 - 3.69i)T + 47iT^{2} \)
53 \( 1 + (-2.83 + 2.83i)T - 53iT^{2} \)
59 \( 1 - 5.54T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 + (3.75 - 3.75i)T - 67iT^{2} \)
71 \( 1 - 3.61iT - 71T^{2} \)
73 \( 1 + (5.89 + 5.89i)T + 73iT^{2} \)
79 \( 1 + 17.0iT - 79T^{2} \)
83 \( 1 + (-3.21 + 3.21i)T - 83iT^{2} \)
89 \( 1 + 9.40T + 89T^{2} \)
97 \( 1 + (-4.39 + 4.39i)T - 97iT^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.20022062384677953489755988403, −12.29702958566334132957661643588, −11.36723517438981323741127416896, −10.44584963290019825794773748428, −9.861633913763590179554505186327, −8.877151148657769709545239781209, −7.20648096252007452394109509353, −5.83178422810572897675169052316, −3.68148429281473935071018940790, −1.77910111275083157188588187375, 0.993962823957079229028518228624, 5.25109793086912669944278656656, 5.97561854283333717619995786257, 7.04691426356354335057867992346, 8.180644809745310338857011068648, 9.446390035966923783393625046059, 10.19469734967892822148452199455, 11.32275138446440083219282456050, 12.95656005229518368560592341832, 13.84399388538716480163770313995

Graph of the $Z$-function along the critical line